1. Field of the Invention
The present invention relates to the field of quantum information processing such as quantum computation, quantum communication, and quantum cryptography, and more specifically relates to an apparatus for generating an entangled quantum state of a plurality of qubits (two-state quantum systems), an apparatus for performing Bell measurement, which is simultaneous measurement on a two-qubit system, an apparatus for implementing a controlled-NOT gate, which is unitary transformation of a two-qubit system, and a method for performing approximate evaluation of the fidelity of a quantum gate composed of an interferometer which performs an interaction-free measurement (IFM) (hereafter called an IFM interferometer).
2. Description of the Related Art
Since it was found that quantum computation can solve some kinds of problems more efficiently than classical computers, quantum computation has been extensively researched (see references (1) to (4)). In addition, as research on quantum information processing, such as quantum teleportation, has become popular, the importance of physical phenomenon called quantum entanglement has become more recognized (see references (5) and (6)).
Quantum computation is performed by preparing a plurality of two-state quantum systems called qubits and successively performing unitary transformation of them and observing the results. Any unitary transformation applied to the qubits can be decomposed into U(2) transformations applied to a single qubit and controlled-NOT gates operating on two qubits (see reference (7)). The controlled-NOT gate produces quantum entanglement between two qubits, and various methods have been proposed and experiments have been carried out to realize a controlled-NOT gate. For example, a method using cavity quantum electrodynamics (QED) (see references (8) and (9)), a method for implementing the operation with a certain probability using linear optical devices (see reference (10)), and a method using a superconductive Josephson junction (see reference (11)) have been proposed. However, all of these methods require highly advanced experimental techniques, and are not expected to be put into practical use in the near future. If a controlled-NOT gate were realized, Bell measurement, which is simultaneous measurement on a two-qubit system, could also be realized.
On the other hand, experimental methods for generating an entangled quantum state is also researched. Quantum entanglement is a quantum-mechanical correlation between two systems which can be locally separated from each other. More specifically, in a state called a pure state in quantum mechanics, if the overall state of two systems A and B cannot be expressed in the form of a simple product |ΨAB>=|ψA>{circle around (x)}|φB>, |ΨAB> is entangled and the system AB is in a state of quantum entanglement. In this state, neither classical communication between the systems A and B nor a local operation in each of the systems A and B (unitary transformation of each of the systems A and B, addition of an auxiliary system, and observation of a local degree of freedom) is possible. Accordingly, it is considered that the entangled state has a correlation which cannot be explained by classical probability theory (see references (12) to (15)).
In a two-qubit system, typical states of quantum entanglement are Bell states, which are expressed as follows:|Φ±>=(1/√{square root over (2)}) (|00>±|11>)|Ψ±>=(1/√{square root over (2)}) (|01>±|10>)  (1.1)where {|0>, |1>} are orthogonal bases of a two-dimensional Hilbert space in which two qubits are defined. In Expression (1.1), {|Φ±>, |Ψ±>} are orthogonal bases of a four-dimensional Hilbert space spanned by the two qubits, and are therefore called Bell bases. The Bell states play an important role in quantum teleportation.
One known method for generating two particles in a Bell state is parametric down-conversion, in which a nonlinear optical crystal such as beta-barium borate (BBO) and LiIO3 is irradiated with ultraviolet pulses so that pair creation of two photons whose polarization degrees of freedom are in a Bell state occurs (see reference (16)). In this method, however, the occurrence rate of the down-conversion is determined by two-dimensional nonlinear susceptibility χ(2), and therefore the generation efficiency of Bell photon pairs (Bell pairs) is low. Accordingly, the intensity of the ultraviolet pulses must be increased in actual experiments. Note that the Bell states can be easily generated in a system where the controlled-NOT gate transformation can be freely implemented. Since it is extremely difficult to realize a controlled-NOT gate, only a method for directly generating the Bell state is described here.
The Bell measurement is simultaneous measurement on a 2-qubit system performed for distinguishing four Bell bases {|Φ±>, |Ψ±>} from one another. In addition to the controlled-NOT gate, the Bell measurement is also a basic operation in quantum information processing, and is essential in quantum teleportation. Gottesman and Chuang have proved that the controlled-NOT gate can be implemented by generating a particular four-qubit state:|χ>=(½)[(|00>+|11>)|00>+(|01>+|10>) |11>]  (1.2)and performing the Bell measurement twice and single-qubit unitary transformations depending on the result of the Bell measurement (see reference (17)).
In the following description, an interaction-free measurement (IMF) is adopted as the fundamental concept. The IFM is an observation method formulated by Elitzur and Vaidman and derived to solve the following problem. That is, “when there is an object which always absorbs a photon by a strong interaction if the photon comes near enough to the object, how can it be decided whether this object is present or absent without causing it to absorb the photon?” The reason why the photon is preferably not absorbed by the object is because, for example, there is a risk that the object will explode if it absorbs the photon.
The means by which Elitzur and Vaidman solved this problem will be described below (see also references (18) and (19)). FIG. 20 is a diagram showing an experiment of an interaction-free measurement (IFM) performed by Elitzur and Vaidman. In this experiment, a Mach-Zehnder interferometer including two beam splitters which act as boundaries between an upper path a and a lower path b is used. A state in which a single photon is present on the path a is expressed as |1>a and a state in which no photon is present on the path a is expressed as and |0>a. In addition, an orthogonal relationship a<i|j>a=δij is satisfied for any i and j (i,jε{0, 1}). These settings are similar for the path b. The operations of the two beam splitters B and B′ are defined as follows:
                    B        :                  {                                                                                                                                                                1                        〉                                            a                                        ⁢                                                                                          0                        〉                                            b                                                        →                                                            cos                      ⁢                                                                                          ⁢                      θ                      ⁢                                                                                                  1                          〉                                                a                                            ⁢                                                                                                  0                          〉                                                b                                                              -                                          sin                      ⁢                                                                                          ⁢                      θ                      ⁢                                                                                                  0                          〉                                                a                                            ⁢                                                                                                  1                          〉                                                b                                                                                                                                                                                                                                                            0                        〉                                            a                                        ⁢                                                                                          1                        〉                                            b                                                        →                                                            sin                      ⁢                                                                                          ⁢                      θ                      ⁢                                                                                                  1                          〉                                                a                                            ⁢                                                                                                  0                          〉                                                b                                                              +                                          cos                      ⁢                                                                                          ⁢                      θ                      ⁢                                                                                                  0                          〉                                                a                                            ⁢                                                                                                  1                          〉                                                b                                                                                                                                                    (        1.3        )                                          B          ′                :                  {                                                                                                                                                                1                        〉                                            a                                        ⁢                                                                                          0                        〉                                            b                                                        →                                                            sin                      ⁢                                                                                          ⁢                      θ                      ⁢                                                                                                  1                          〉                                                a                                            ⁢                                                                                                  0                          〉                                                b                                                              +                                          cos                      ⁢                                                                                          ⁢                      θ                      ⁢                                                                                                  0                          〉                                                a                                            ⁢                                                                                                  1                          〉                                                b                                                                                                                                                                                                                                                            0                        〉                                            a                                        ⁢                                                                                          1                        〉                                            b                                                        →                                                            sin                      ⁢                                                                                          ⁢                      θ                      ⁢                                                                                                  1                          〉                                                a                                            ⁢                                                                                                  0                          〉                                                b                                                              -                                          sin                      ⁢                                                                                          ⁢                      θ                      ⁢                                                                                                  0                          〉                                                a                                            ⁢                                                                                                  1                          〉                                                b                                                                                                                                                    (        1.4        )            The upper path a of the interferometer is placed on a point where the presence/absence of the object is to be determined.
The case in which a photon is injected into the path b from the lower left is considered. When nothing is present on the two paths a and b of the interferometer, the photon comes out from the path a at the upper right and is detected by a detector D0. In comparison, when an object which can absorb the photon is present on the upper path a, the object absorbs the photon with probability 1 if the photon comes near enough to the object to cause the interaction. Accordingly, there are three possibilities:    (A) Neither of detectors D0 and D1 detects the photon: probability PA=sin2 θ    (B) Detector D0 detects the photon: Probability PB=cos4 θ    (C) Detector D1 detects the photon: Probability PC=cos2 θ sin2 θ
(A) means that the photon has been absorbed by the object, and therefore the condition of IFM is not satisfied. In addition, (B) means that the presence/absence of the object cannot be determined. (C) means that the presence of the object is detected without causing the object to absorb the photon. Elitzur and Vaidman called the operation (C) interaction-free measurement. As used here, the term “interaction-free” describes the case where the photon has not been absorbed by the object.
The efficiency ζ of the IFM is calculated as follows:ζ=PC/(PA+PC)  (1.5)The reason why PB is not included in Expression (1.5) is because the experiment can be retried in the case (B). When θ=π/4, the beam splitters B and B′ serve as 50—50 beam splitters (beam splitters whose transmittance and reflectance are both ½), and PA, PB, PC, and ζ are determined as PA=½, PB=PC=¼, and ζ=⅓, respectively. Generally, ζ is calculated as follows:ζ=z/(1+z), z=cos2 θ, 0≦z≦1  (1.6)FIG. 21 shows a graph of the efficiency ζ versus the reflectance z (0≦z≦1) of the beam splitters, and it is clear from this graph that ζ≦½.
Accordingly, in the method according to Elitzur and Vaidman, the efficiency ζ never exceeds ½. In addition, PB approaches 1 as ζ approaches ½, which means that the number of retries increases. When the object is present on the path a, the average number of tries taken until the measurement finishes by obtaining the result (A) or (C) is calculated as N=1/(1−PB)=1/(1−cos4 θ). Accordingly, N diverges to infinity ( N→∞) when ζ→½ or θ→0. In other words, the number of tries diverges to infinity as ζ approaches ½.
Kwiat et al. have created a method for causing ζ to asymptotically approach 1 and PB to asymptotically approach 0 (see references (20) and (21)). In the method according to Kwiat et al., an interferometer shown in FIG. 22 is used, which includes N beam splitters which act as boundaries between an upper path a and a lower path b. Similar to the above-described case, a state in which a single photon is present on the path a is expressed as |1>a and a state in which no photon is present on the path a is expressed as |0>a. In addition, these settings are similar for the path b. The operations of the beam splitters B are defined by Expression (1.3).
A photon is injected through the lower left entrance b. When nothing is present on the paths, the wave function of the photon which comes out from the kth beam splitter is expressed as follows:sin kθ|1>a|0>b+cos kθ|0>a|1>b, k=0, 1 . . . , N  (1.7)When θ=π/2N, the photon comes out from the upper right exit a of the Nth beam splitter with probability 1.
Next, the case is considered in which N identical objects which can absorb the photon are present on the upper path a at positions behind the beam splitters. In this case, the photon injected through the lower left entrance b cannot pass through the path a since it will be absorbed by the objects if it enters the path a. Accordingly, the probability P that the photon will come out from the lower right exit b is calculated as the product of the reflectances of the beam splitters (P=cos2N θ) When N increases to infinity, P approaches 1:
                                          lim                          N              →              ∞                                ⁢                                          ⁢          P                =                                            lim                              N                →                ∞                                      ⁢                                                  ⁢                                          cos                                  2                  ⁢                  N                                            ⁡                              (                                  π                                      2                    ⁢                    N                                                  )                                              =                                                    lim                                  N                  →                  ∞                                            ⁢                              [                                  1                  -                                                            π                      2                                                              4                      ⁢                      N                                                        +                                      O                    ⁡                                          (                                              1                                                  N                          2                                                                    )                                                                      ]                                      =            1                                              (        1.8        )            Accordingly, the efficiency ζ (=P) in detecting the objects by the IFM approaches 1 when N→∞.
As is clear from the above-discussion, the interferometer according to Kwiat et al. changes the direction in which the photon injected from the lower left travels as follows, at least with probability P:    (1) If no absorbing object is present in the interferometer, the photon comes out from the upper right exit a.    (2) If the absorbing objects are present in the interferometer, the photon will come out from the lower right exit b.In addition, P approaches 1 as N increases. In the following description, the interferometer shown in FIG. 22 proposed by Kwiat et al. is called an IFM interferometer.
The documents listed below are incorporated herein by reference:    (1) D. Deutsch and R. Jozsa, “Rapid solution of problems by quantum computation”, Proc. R. Soc. London, Ser. A 439, 553–558 (1992).    (2) D. R. Simon, “On the power of quantum computation”, SIAM J. Comput. 26, 1474–1483 (1997).    (3) P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer”, SIAM J. Comput. 26, 1484–1509 (1997).    (4) L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack”, Phys. Rev. Lett. 79, 325–328 (1997).    (5) C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels”, Phys. Rev. Lett. 70, 1895–1899 (1993).    (6) D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation”, Nature (London) 390, 575–579 (1997).    (7) A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, “Elementary gates for quantum computation”, Phys. Rev. A 52, 3457–3467 (1995).    (8) Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic”, Phys. Rev. Lett. 75, 4710–4713 (1995).    (9) C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, “Demonstration of a fundamental quantum logic gate”, Phys. Rev. Lett. 75, 4714–4717 (1995).    (10) E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics”, Nature (London) 409, 46–52 (2001).    (11) T. Yamamoto, Yu. A. Pashkin, 0. Astafiev, Y. Nakamura, and J. S. Tsuai, “Demonstration of conditional gate operation using superconducting charge qubits”, Nature (London) 425, 941–944 (2003).    (12) J. S. Bell, “Speakable and unspeakable in quantum mechanics” (Oxford, Oxford University Press, 1983).    (13) C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction”, Phys. Rev. A 54, 3824–3851 (1996).    (14) R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model”, Phys. Rev. A 40, 4277–4281 (1989).    (15) S. Popescu, “Bell's inequalities and density matrices: revealing “hidden” nonlocality”, Phys. Rev. Lett. 74, 2619–2622 (1995).    (16) P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs”, Phys. Rev. Lett. 75, 4337–4341 (1995).    (17) D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations”, Nature (London) 402, 390–393 (1999).    (18) A. C. Elitzur and L. Vaidman, “Quantum mechanical interaction-free measurements”, Found. Phys. 23, 987–997 (1993).    (19) L. Vaidman, “Are interaction-free measurements interaction free?”, Opt. Spectrosc. 91, 352–357 (2001).    (20) P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich, “Interaction-free measurement”, Phys. Rev. Lett. 74, 4763–4766 (1995).    (21) P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nariz, G. Weihs, H. Weinfurter, and A. Zeilinger, “High-efficiency quantum interrogation measurements via the quantum Zeno effect”, Phys. Rev. Lett. 83, 4725–4728 (1999).
In the field of quantum information processing such as quantum computing, quantum communication, and quantum cryptography, the three most important basic operations are: generation of the Bell state, the Bell measurement, and the controlled-NOT gate transformation. These three operations are not independent from one another but are closely related to one another, and there are demands for these operations.
As described above, one known method for generating two particles in a Bell state is parametric down-conversion, in which a nonlinear optical crystal is irradiated with ultraviolet pulses so that pair creation of two photons whose polarization degrees of freedom are in a Bell state occurs. In this method, however, the occurrence rate of the down-conversion is determined by the two-dimensional nonlinear susceptibility χ(2), and therefore the generation efficiency of Bell photon pairs is low. Accordingly, the intensity of the ultraviolet pulses must be increased in actual experiments.
The Bell measurement is an essential technique in quantum teleportation, and it is an important objective in quantum information processing to create a simple method for the Bell measurement.
The controlled-NOT gate is regarded as an essential technique to realize a quantum computer. More specifically, a U(2) transformation gate for a single qubit and a controlled-NOT gate form a universal set of gates for quantum computation, and it is known that any kind of operation on qubits can be implemented by combining these gates. Accordingly, to realize a controlled-NOT gate is one of the most important objectives in quantum information processing. If a controlled-NOT gate were realized, generation of the Bell state and the Bell measurement could also be realized.
However, it is difficult to realize a controlled-NOT gate which produces a quantum correlation between two qubits. Although a method using cavity QED and other methods have been proposed, as described above, these methods require highly advanced experimental techniques, and are not expected to be put into practical use in the near future.
On the other hand, Gottesman and Chuang have proved that a controlled-NOT gate can be implemented by generating a particular four-qubit state |χ> and performing the Bell measurement twice and single-qubit unitary transformations depending on the result of the Bell measurement. This means that although it is difficult to implement the controlled-NOT gate directly, it can be implemented indirectly if the Bell measurement can be performed easily.
Thus, the Bell measurement and the controlled-NOT gate are closely related to each other, and there is a requirement to create a simple method for the Bell measurement and to thereby realize a controlled-NOT gate.